ABSTRACT
In this section we take up the question of representing an endomorphism of a finite-dimensional vector space by a matrix of a form which is as simple as possible. In the last section we saw that every endomorphism which has all its eigenvalues in the given base-field can be represented by a triangular matrix. However, an endomorphism can usually be represented by many different triangular matrices, and it would be nice if we could specify a particularly simple form to represent such an endomorphism. To get an idea of how such a “canonical form” for an arbitrary endomorphism can be obtained, we observe that the diagonalizability of an endomorphism T: V → V can be expressed by saying that V = ker(T − λ 1 1) ⊕ ⋯ ⊕ ker(T − λ k 1) where λ 1,…,λ k , are the eigenvalues of T. For example, the diagonalizability results (6.27) and (6.28) were obtained by realizing that V = ker(P) ⊕ ker(P − 1) for a projection P and that V = ker(T − 1) ⊕ ker(T + 1) for a reflection T. We will see that for an arbitrary endomorphism A: V → V there is a decomposition V = ker p 1(A) ⊕ ⋯ ⊕ ker p n (A) where each p i is a polynomial. Therefore, we are going to study polynomial expressions a 0 1 + a 1 A + ⋯ + a n A n of an endomorphism A, thereby exploiting the fact that Hom(V, V) is not merely a vector space but is an algebra in the sense of the following definition.
